#211802

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄×#121
prime radical	#19133	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,18,22,16)(11,19,12,15,20,24,21,23);(9,11,13,12,14,20,22,21)(10,15,16,19,18,23,17,24);(2,5)(3,8)(6,7)(11,20)(12,21)(15,23)(19,24);(9,12,13,20,14,21,22,11)(10,15,17,24,18,23,16,19)

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=dh,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118998);